Theorem ssmapsn 38403

Description: A subset 𝐶 of a set exponentiation to a singleton, is its projection 𝐷 exponentiated to the singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
ssmapsn.f	⊢ Ⅎ𝑓𝐷
ssmapsn.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ssmapsn.c	⊢ (𝜑 → 𝐶 ⊆ (𝐵 ↑𝑚 {𝐴}))
ssmapsn.d	⊢ 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓

Assertion

Ref	Expression
ssmapsn	⊢ (𝜑 → 𝐶 = (𝐷 ↑𝑚 {𝐴}))

Distinct variable groups:   𝐴,𝑓   𝐶,𝑓   𝜑,𝑓

Allowed substitution hints:   𝐵(𝑓)   𝐷(𝑓)   𝑉(𝑓)

Proof of Theorem ssmapsn

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssmapsn.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ (𝐵 ↑𝑚 {𝐴}))
2	1	sselda 3568	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐵 ↑𝑚 {𝐴}))
3		elmapi 7765	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐵 ↑𝑚 {𝐴}) → 𝑓:{𝐴}⟶𝐵)
4	2, 3	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓:{𝐴}⟶𝐵)
5	4	ffnd 5959	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 Fn {𝐴})
6		ssmapsn.d	. . . . . . . . 9 ⊢ 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓
7	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓)
8		ovex 6577	. . . . . . . . . . . 12 ⊢ (𝐵 ↑𝑚 {𝐴}) ∈ V
9	8	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ↑𝑚 {𝐴}) ∈ V)
10	9, 1	ssexd 4733	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ V)
11		rnexg 6990	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝐶 → ran 𝑓 ∈ V)
12	11	rgen 2906	. . . . . . . . . . 11 ⊢ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V
13	12	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V)
14	10, 13	jca 553	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ V ∧ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V))
15		iunexg 7035	. . . . . . . . 9 ⊢ ((𝐶 ∈ V ∧ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V)
16	14, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V)
17	7, 16	eqeltrd 2688	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ V)
18	17	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝐷 ∈ V)
19		ssiun2 4499	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐶 → ran 𝑓 ⊆ ∪ 𝑓 ∈ 𝐶 ran 𝑓)
20	19	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → ran 𝑓 ⊆ ∪ 𝑓 ∈ 𝐶 ran 𝑓)
21		ssmapsn.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
22		snidg 4153	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
23	21, 22	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
24	23	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝐴 ∈ {𝐴})
25		fnfvelrn 6264	. . . . . . . . 9 ⊢ ((𝑓 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → (𝑓‘𝐴) ∈ ran 𝑓)
26	5, 24, 25	syl2anc 691	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ran 𝑓)
27	20, 26	sseldd 3569	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ∪ 𝑓 ∈ 𝐶 ran 𝑓)
28	27, 6	syl6eleqr 2699	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ 𝐷)
29	5, 18, 28	elmapsnd 38391	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐷 ↑𝑚 {𝐴}))
30	29	ex 449	. . . 4 ⊢ (𝜑 → (𝑓 ∈ 𝐶 → 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})))
31	17	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) → 𝐷 ∈ V)
32		snex 4835	. . . . . . . . . 10 ⊢ {𝐴} ∈ V
33	32	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) → {𝐴} ∈ V)
34		simpr 476	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) → 𝑓 ∈ (𝐷 ↑𝑚 {𝐴}))
35	23	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) → 𝐴 ∈ {𝐴})
36	31, 33, 34, 35	fvmap 38382	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) → (𝑓‘𝐴) ∈ 𝐷)
37	6	idi 2	. . . . . . . . 9 ⊢ 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓
38		rneq 5272	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
39	38	cbviunv 4495	. . . . . . . . 9 ⊢ ∪ 𝑓 ∈ 𝐶 ran 𝑓 = ∪ 𝑔 ∈ 𝐶 ran 𝑔
40	37, 39	eqtri 2632	. . . . . . . 8 ⊢ 𝐷 = ∪ 𝑔 ∈ 𝐶 ran 𝑔
41	36, 40	syl6eleq 2698	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) → (𝑓‘𝐴) ∈ ∪ 𝑔 ∈ 𝐶 ran 𝑔)
42		eliun 4460	. . . . . . 7 ⊢ ((𝑓‘𝐴) ∈ ∪ 𝑔 ∈ 𝐶 ran 𝑔 ↔ ∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔)
43	41, 42	sylib 207	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) → ∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔)
44		simp3 1056	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → (𝑓‘𝐴) ∈ ran 𝑔)
45		simp1l 1078	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝜑)
46	45, 21	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝐴 ∈ 𝑉)
47		eqid 2610	. . . . . . . . . . 11 ⊢ {𝐴} = {𝐴}
48		simp1r 1079	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷 ↑𝑚 {𝐴}))
49		elmapfn 7766	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐷 ↑𝑚 {𝐴}) → 𝑓 Fn {𝐴})
50	48, 49	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴})
51	1	sselda 3568	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 ∈ (𝐵 ↑𝑚 {𝐴}))
52		elmapfn 7766	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝐵 ↑𝑚 {𝐴}) → 𝑔 Fn {𝐴})
53	51, 52	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 Fn {𝐴})
54	53	3adant3 1074	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
55	54	3adant1r 1311	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
56	46, 47, 50, 55	fsneqrn 38398	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → (𝑓 = 𝑔 ↔ (𝑓‘𝐴) ∈ ran 𝑔))
57	44, 56	mpbird 246	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 = 𝑔)
58		simp2 1055	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 ∈ 𝐶)
59	57, 58	eqeltrd 2688	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ 𝐶)
60	59	3exp 1256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) → (𝑔 ∈ 𝐶 → ((𝑓‘𝐴) ∈ ran 𝑔 → 𝑓 ∈ 𝐶)))
61	60	rexlimdv 3012	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) → (∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔 → 𝑓 ∈ 𝐶))
62	43, 61	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})) → 𝑓 ∈ 𝐶)
63	62	ex 449	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐷 ↑𝑚 {𝐴}) → 𝑓 ∈ 𝐶))
64	30, 63	impbid 201	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})))
65	64	alrimiv 1842	. 2 ⊢ (𝜑 → ∀𝑓(𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})))
66		nfcv 2751	. . 3 ⊢ Ⅎ𝑓𝐶
67		ssmapsn.f	. . . 4 ⊢ Ⅎ𝑓𝐷
68		nfcv 2751	. . . 4 ⊢ Ⅎ𝑓 ↑𝑚
69		nfcv 2751	. . . 4 ⊢ Ⅎ𝑓{𝐴}
70	67, 68, 69	nfov 6575	. . 3 ⊢ Ⅎ𝑓(𝐷 ↑𝑚 {𝐴})
71	66, 70	dfcleqf 38281	. 2 ⊢ (𝐶 = (𝐷 ↑𝑚 {𝐴}) ↔ ∀𝑓(𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑𝑚 {𝐴})))
72	65, 71	sylibr 223	1 ⊢ (𝜑 → 𝐶 = (𝐷 ↑𝑚 {𝐴}))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125  ∪ ciun 4455  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746

This theorem is referenced by:  vonvolmbllem  39550  vonvolmbl2  39553  vonvol2  39554